Conic Section Exercises
After an earthquake, three seismic stations record the wave at the same time. How do you locate the epicenter? You already know the four conic sections separately. Now the challenge is to recognize them quickly, either from their equation or from the condition their points satisfy, and to solve the whole problem. At the end of the page you come back to this earthquake.
Which conic is it? Look at the equation
To find the center, vertices, and foci you have to complete the square, just as with special products.
Determine the coordinates of the center and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Determine the coordinates of the center and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Given the conic with general equation $9x^2 + 25y^2 - 54x + 200y + 256 = 0$, classify it by completing the square, and find the coordinates of its center, its vertices,…
Given the conic with general equation $9x^2 + 25y^2 - 54x + 200y + 256 = 0$, classify it by completing the square, and find the coordinates of its center, its vertices, its foci, and the value of its eccentricity.
Which conic is it? Look at the statement
When there is no equation, look at the condition that the points $P(x,y)$ satisfy:
In the last row, if the ratio $e$ is less than $1$ the result is an ellipse, if it is $1$ a parabola, and if it is greater than $1$ a hyperbola.
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its…
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its standard equation.
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the…
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's path, simplify it, and identify the type of geometric curve it describes.
Perpendicular bisector and angle bisectors
Two simple loci always show up in problems.
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Perpendicular bisector of $AB$: points at the same distance from $A$ and from $B$. It is the line perpendicular to $AB$ that passes through its midpoint.
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Angle bisectors of two lines: points at the same distance from both lines.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Loci with slopes or with a directrix
Two more situations produce conic sections:
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The product of the slopes of $PA$ and $PB$ is constant.
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The quotient between the distance to a point (focus) and to a line (directrix) is constant, equal to $e$.
The method is always the same: write the condition with $x$ and $y$, clear denominators or roots, and simplify.
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is…
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is equal to $-\frac{9}{16}$. Identify the resulting conic, calculate its eccentricity, and sketch it.
Find the equation of the locus of points in the plane whose distance from the fixed point $F(4, 0)$ is equal to half their distance from the vertical line $x = 16$.…
Find the equation of the locus of points in the plane whose distance from the fixed point $F(4, 0)$ is equal to half their distance from the vertical line $x = 16$. Identify the resulting conic, find its semi-axes, and sketch it, illustrating the directrix property.
Circle: center, radius, three points, and distance to a line
Two forms of the equation:
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Standard: $(x-h)^2+(y-k)^2=r^2$, with center $(h,k)$ and radius $r$.
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General: $x^2+y^2+Dx+Ey+F=0$, with center $\left(-\tfrac D2,-\tfrac E2\right)$ and $r^2=h^2+k^2-F$.
If you are given three points, substitute each one into the general form. You get three equations with $D$, $E$, and $F$. Another option is to intersect two perpendicular bisectors: the point where they cross is the center, which is at the same distance from the three points.
A circle has its center at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
A circle has its center at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometers). Assuming that the seismic waves travel at the same speed…
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometers). Assuming that the seismic waves travel at the same speed and arrive at the same time at all three stations, find the coordinates of the epicenter and the radius of the affected area (distance to the stations).
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation…
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation $3x + 4y + k = 0$. Determine the values of the parameter $k$ for which the ship is detected by the radar at some point along its path.
Ellipse: from the description to the equation
With the center at the origin and a horizontal major axis, $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ with $a>b>0$. We have $c^2=a^2-b^2$, the foci are $(\pm c,0)$, and $e=\dfrac ca<1$. If the major axis is vertical, $a^2$ goes under $y^2$ and the foci are $(0,\pm c)$.
Find the lengths of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
Find the lengths of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is…
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is $3 \text{ Mm}$ and the maximum distance (apoapsis) is $9 \text{ Mm}$. Determine the equation of the orbit and its eccentricity, assuming that the center of the ellipse is at the origin and the focal axis lies on the x-axis.
Hyperbola: branches, asymptotes, and the rectangular hyperbola
With the center at the origin and a horizontal transverse axis, $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Here $c^2=a^2+b^2$, the vertices are $(\pm a,0)$, the foci $(\pm c,0)$, the asymptotes $y=\pm\dfrac bax$, and $e=\dfrac ca>1$. In the ellipse $a$ is the larger one; in the hyperbola, $a^2$ is simply the denominator of the positive term.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is at a height of $50 \text{ m}$…
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is at a height of $50 \text{ m}$ above the ground. If the base of the tower has a width of $80 \text{ m}$, find the equation of the hyperbola, assuming its center coincides with the center of the narrowest part.
A subatomic particle moves in a magnetic field, describing an equilateral hyperbolic path whose asymptotes coincide with the coordinate axes. Knowing that the particle…
A subatomic particle moves in a magnetic field, describing an equilateral hyperbolic path whose asymptotes coincide with the coordinate axes. Knowing that the particle passes through the point $P(4, 2)$, solve:
- a) Find the equation of the path.
- b) Determine the exact coordinates of the vertices and the foci.
- c) Sketch the curve (the figure shows the axes and the point $P$; you can copy it into your notebook).
You can do it on paper and send a photo with your answer.
Parabola in context problems
With the vertex at the origin:
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$y^2=4px$ (opens to the right if $p>0$, to the left if $p<0$): focus $(p,0)$, directrix $x=-p$.
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$x^2=4py$ (opens upward if $p>0$, downward if $p<0$): focus $(0,p)$, directrix $y=-p$.
The latus rectum has length $|4p|$. Change $p$ and watch how the opening changes:
In context problems, the hard part is placing the axes so that the vertex is easy to locate.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck.…
The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck. The lowest point of the cable is $10 \text{ m}$ above the deck. Determine the equation of the parabola and calculate the height of the cable $30 \text{ m}$ from the center of the bridge.
Conics and lines: intersection points and tangents
To find the intersection points, solve for one variable from the line and substitute it into the conic. You get a quadratic equation whose discriminant $\Delta$ decides: $\Delta>0$, two points (secant); $\Delta=0$, one (tangent); $\Delta<0$, none.
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has x-coordinate $x = 3$ and…
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has x-coordinate $x = 3$ and a positive y-coordinate. Find the equation of the tangent line to the mirror at that point.
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of…
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of intersection.
You can do it on paper and send a photo with your answer.
A comet follows a parabolic path given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact…
A comet follows a parabolic path given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact coordinates of the points of intersection of both paths.
Tangents from a point, angles, and the radical axis
Tangents from an exterior point
Write the line through the point with slope $m$ and require that its distance to the center equal the radius. You solve an equation in $m$ (if there is a vertical line, check it separately).
Angle between two lines
If the slopes are $m_1$ and $m_2$, then $\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|$. For example, with $m_1=3$ and $m_2=\tfrac12$: $\tan\theta=\left|\dfrac{-5/2}{5/2}\right|=1$, so $\theta=45^\circ$. This is useful for seeing how a ray reflects off a parabola: the angle between the ray and the tangent is the same as between the tangent and the reflected ray, which leaves parallel to the axis.
Radical axis of two circles
If you subtract the general equations of two circles, the $x^2$ and $y^2$ terms cancel and a line remains: the radical axis. If the circles intersect, that line passes through the two intersection points.
From a control station located at point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation…
From a control station located at point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation $x^2 + y^2 - 4x + 2y - 15 = 0$. Find the equations of the two straight-line paths tangent to the orbit from the control station.
A car headlight has a parabolic cross-section. The bulb is located at the focus $F(3, 0)$ and the vertex of the parabola is at the origin. A ray of light leaves the…
A car headlight has a parabolic cross-section. The bulb is located at the focus $F(3, 0)$ and the vertex of the parabola is at the origin. A ray of light leaves the bulb, strikes the parabola at a point with y-coordinate $y = 6$, and is reflected. Calculate the equation of the tangent line at the point of incidence and find the angle that the incident ray makes with this tangent.
Two telecommunications networks have coverage areas modelled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of…
Two telecommunications networks have coverage areas modelled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of the radical axis, prove analytically that the circles intersect at two points, and calculate the exact distance between their points of intersection.